More than 6,500 people registered to vote in New Hampshire on Nov. 8 using out-of-state driver’s licenses, and since then the vast majority have neither obtained an in-state license nor registered a motor vehicle.

Conservatives say the state’s same-day registration is an invitation for fraud because of loose proof-of-residence rules.

New Hampshire House Speaker Shawn Jasper, a Republican, issued the numbers Thursday based on inquiries he made to the Department of State, which oversees elections, and the Department of Safety.

Since Election Day, Republicans have charged that a significant number of nonresident Democrats, principally from Massachusetts, flowed into New Hampshire to vote illegally, tilting a close race to their party.

“Having worked before on a campaign in New Hampshire, I can tell you that this issue of busing voters into New Hampshire is widely known by anyone who’s worked in New Hampshire politics. It’s very real. It’s very serious. This morning, on this show, is not the venue for me to lay out all the evidence,” White House policy adviser Stephen Miller told ABC News in February.

Though Mr. Jasper’s findings don’t prove those accusations, they do corroborate them. The numbers read this way:

Logan Churchwell, spokesman for Public Interest Legal Foundation, which investigates voter fraud, said Mr. Jasper’s numbers bolster his group’s findings that many people vote in New Hampshire without proof of residence.

“We’ve known for months that more voters cast ballots without any proof of actually living in New Hampshire than the differentials for either federal contest there in 2016,” he said. “Now it looks like they were back in Boston in time to watch the election returns that evening. The left-wing groups suing to block new proof-of-residence laws for same-day voter registration are really proving what drives them to the courthouse.”

Project Veritas, a conservative investigative unit, took hidden cameras to New Hampshire for the 2016 February primary.

Poll workers told Veritas operatives that they did not need to live in the state to vote, that they could use a Massachusetts driver’s license and that they could fill out a form if they had no ID.

“If you’re here today, you can vote and be gone,” one poll official unwittingly told Veritas.

A spokesman for Mr. Jasper said the speaker was presenting raw data and did not know which states issued the 6,540 licenses and acknowledged that the numbers could include some college students.

In February, while meeting with senators at the White House, Mr. Trump said he lost New Hampshire because thousands of Massachusetts residents crossed state lines to vote. He also said Ms. Ayotte lost for the same reason: illegal voting.

The liberal media dismissed his accusations. The Boston Globe called them “groundless.”

The president has appointed a special commission to investigate voter integrity, led by Vice President Mike Pence.

By coincidence, New Hampshire will host the Presidential Advisory Commission on Election Integrity’s first meeting outside Washington on Tuesday. Vice Chairman Kris Kobach, the Kansas secretary of state, will preside.

Democrats oppose the panel and have called for its demise.

At least two scientific surveys show that a larger number of noncitizens register and vote illegally in U.S. elections. One poll found that a large majority vote Democrat.

Mr. Jasper said in a statement that he requested the driver’s license and motor vehicle information “to benefit the legislature in its assessment of the effectiveness of our current election laws as well as future legislation that could improve our voter registration and verification processes.”

The two state departments, State and Safety, provided the data to Mr. Jasper on Wednesday in a joint letter.

Secretary of State William M. Gardner, a Democrat, signed the letter. Also signing was John Barthelmes, the Republican appointed commissioner for the Department of Safety.

The two agencies explained the 5,313 number (neither a driver’s license nor a registered motor vehicle many months later) with several possible reasons.

“It is likely that some unknown number of these individuals moved out of New Hampshire, it is possible that a few may have never driven in New Hampshire or have ceased driving, however, it is expected that an unknown number of the remainder continue to live and drive in New Hampshire. If they have established their residence in New Hampshire, they may have failed to obtain a New Hampshire driver’s license.”

Nowhere in the letter do the departments suggest that out-of-state people voted illegally.

The letter contains another interesting fact: There were 6,033 people who signed domicile affidavits in order to register to vote. The secretary of state sent letters to all of them, and the U.S. Postal Service returned 458 of them as not delivered to the address on the registration forms.

“Using cherry-picked data in order to support a false claim is dangerous and irresponsible. Today’s release of information by Speaker Jasper’s office fans the flames of misinformation in order to further suppress our citizens’ right to vote,” said Senate Democratic Leader Jeff Woodburn. “The fact of the matter is that there is no requirement to obtain a [New Hampshire] driver’s license or register your car in order to exercise your constitutionally-protected right to vote.”